PINNs Method for Sloving the Probability Response of the Stochastic Linear System with Fractional Gaussian Noise

Abstract In this paper, the Physics-informed neural networks (PINNs) algorithm is mainly used to solve the time-dependent Fokker-Planck-Kolmogorov (FPK) equation, and eventually obtain the transient probability density function (PDF). Firstly, we derive the FPK equation for a dynamical system driven by multiplicative fractional Gaussian noise (fGn). Secondly, a deep learning method based on PINNs is introduced for resolving the corresponding time-dependent FPK equation. Finally, one examples under fGn excitation condition is discussed to determine the effectiveness and feasibility of PINNs algorithm. The findings indicate that PINNs algorithm can get the transient solution of the system under multiplicative fGn. Concurrently, using Monte Carlo method technology is employed to assess the accuracy and calculation efficiency of PINNs algorithm, we find good agreement between the different comparison results, confirming that the PINNs algorithm is not only computationally efficient, but also has good validity and interpretability.